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LUND UNIVERSITY
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Modeling and Optimization of Grade Changes for a Polyethylene Reactor
Larsson, Per-Ola; Andersson, Niklas; Åkesson, Johan; Haugwitz, Staffan
2010
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Citation for published version (APA):Larsson, P-O., Andersson, N., Åkesson, J., & Haugwitz, S. (2010). Modeling and Optimization of GradeChanges for a Polyethylene Reactor. Paper presented at Reglermöte 2010, Lund, Sweden.
Total number of authors:4
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Modeling and Optimization of Grade
Changes for a Polyethylene Reactor
Per-Ola Larsson ∗ Niklas Andersson ∗∗ Johan Akesson ∗
Staffan Haugwitz ∗∗∗
∗ Department of Automatic Control, Lund University, Lund, Sweden(e-mail: Per-Ola.Larsson|[email protected])
∗∗ Department of Chemical Engineering, Lund University, Lund, Sweden(e-mail: [email protected])
∗∗∗ Borealis AB, Stenungsund, Sweden.(e-mail: [email protected])
1. INTRODUCTION
In polymer industry, as in many other industries, thereexist a problem of increasing the capital productivity whilethe market competition has increased the last decades.Manufacturers are forced to switch between different poly-mer grades to suit market demand, that is, runningproduct campaigns on the same plant. It is importantthat the transitions are made in a way such that off-specification polymer is minimized at a low economicalcost, van Brempt et al. (2004).
In this paper we will, using the Modelica and Optimicalanguage, optimize a grade change for a polyethylenereactor.
2. MATHEMATICAL PLANT MODEL
The reactor considered is the first reactor in a Borstar R©
process, see Figure 1. Modeling a reactor is a task includ-ing both theoretical, but also empirical, challenges andhas been accomplished at Borealis AB. Inputs used inthe model are flows of propane, ethylene, and hydrogen,while several outputs such as masses, concentrations anddensities are available as outputs.
The main equations of a reactor model includes materialbalances and non-linear equations for reaction kinetics,catalyst properties, and densities. The resulting modelused for the grade change optimization can be written inthe general non-linear differential algebraic form
0 = F (x, x, w, u)
y = g(x, w, u),(1)
where x is the internal state, w algebraic variables, uinputs, and y outputs.
Pre-poly.
Loop GPR
Catalyst
Monomer
Comonomer
Diluent
Gas
Flash
Polymer Product
outlet
Monomer
Comonomer
Diluent
Fig. 1. Reactor chain of a Borstar R© process
⋆ Sponsored by the Swedish Foundation of Strategic Research in theframework of Process Industry Centre at Lund University (PICLU).
3. MODELING LANGUAGES AND TOOLS
The Modelica modeling language is used to express themathematical model of the reactor. Text-book style declar-ative equations can be expressed as well as acausal com-ponent connections representing physical interfaces, anddifferential and algebraic equations may be mixed resultingin differential algebraic equations. In order to strengthenthe optimization capabilities of Modelica, the Optimica ex-tension has been proposed, see Akesson (2008). Optimicaadds to Modelica a small number of constructs, enablingthe user to conveniently specify dynamic optimizationproblems based on Modelica models.
JModelica.org is a novel Modelica-based open sourceproject targeted at dynamic optimization, see Akessonet al. (2009), and features compilers supporting codegeneration of Modelica/Optimica models to C, a C APIfor evaluating model equations and their derivatives andoptimization algorithms. The platform contains an imple-mentation of a simultaneous optimization method basedon orthogonal collocation on finite elements, see Biegleret al. (2002).
4. OPTIMAL GRADE TRANSITION
The grade transition considered includes three main ob-jectives,
(1) Increase production rate yr by 20%.(2) Increase hydrogen-ethylene ratio yhe by 20%.(3) Keep amount of polymer in reactor ys within specified
bounds during grade transition.
which should be met using the following control variables,
up = propane input flow
ue = ethylene input flow
uh = hydrogen input flow.
Constraints on the inflows and their derivatives are set dueto limitations on actuators and safety precautions.
The first step in the grade change solution procedure isto find the two stationary points corresponding to the twogrades. These are initial value problems of the DAE, i.e.,for t = t0 and t = tf the system fulfills the static equations
0 = F (0, x◦, w◦, u◦)
y◦ = g(x◦, w◦, u◦),(2)
for different values of the variables. The resulting NLPprogram contains approximately around 200 variables andis solved rapidly in the JModelica.org framework.
0 0.2 0.4 0.6 0.8 1
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1
2
4
6
uref
p
up
ue
uref
e
uh
uref
h
Fig. 2. Scaled optimal input flows at grade transition –propane up, ethylene ue and hydrogen uh.
In the dynamic optimization problem, the optimal transi-tion trajectories between the two grades are found.
A quadratic cost function is constructed that includesdeviations from the end grade specifications and inflows,and also inflow derivatives giving an option to control thesmoothness of the input signals. With the reference vectors
yref =[
yref
s yref
r yref
he
]T, uref =
[
uref
p uref
e uref
h
]T,
comprising the stationary solution of the end grade, andthe diagonal weighting matrices Q∆y, Q∆u and Qu, theoptimal control problem representing the transition canbe formulated as
minup,ue,uh
=
tf∫
t0
[
∆y∆uu
]T [
Q∆y 0 00 Q∆u 00 0 Qu
][
∆y∆uu
]
dt (3)
subj. to 0 = F (x, x, w, u)
y = g(x, w, u)
ymin ≤ y ≤ ymax
umin ≤ u ≤ umax,
where the deviation vectors are defined as
∆y = y − yref, ∆u = u − uref,
and the initial values at time t = t0 are defined by thesteady state solution in Eq. (2).
Using the JModelica.org framework and the Modelicamodel resulted in an NLP problem containing approx-imately 20.000–40.000 variables depending on colloca-tion point selection. Using an Intel R© CoreTM2 [email protected], a solution is obtained in about 5–30 min-utes depending on number of variables and initial values.
Figures 2–4 show the resulting optimal trajectories forthe grade transition problem. Note that the trajectorieshave been scaled, either such that initial value is 1 orby constraint. Both inflows u and controlled outputs ytend to the desired reference values specified by the staticoptimization problem while all constraints are fulfilled.
5. FUTURE WORK
Due to the modularity of the Modelica language, theoptimization problem can easily be extended to the fullBorstar R© process. Successfull attempts at performing a
0 0.2 0.4 0.6 0.8 1−1
0
1
0 0.2 0.4 0.6 0.8 1−1
0
1
0 0.2 0.4 0.6 0.8 1−1
0
1
up
ulimit
p
ue
ulimit
e
uh
ulimit
h
Fig. 3. Scaled time derivatives of optimal input flowsat grade transition – propane up, ethylene ue andhydrogen uh.
0 0.2 0.4 0.6 0.8 1
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1
1
1.1
1.2
0 0.2 0.4 0.6 0.8 1
0.95
1
1.05
ys
ylimit
s
yref
s
yr
yref
r
yhe
yref
he
Fig. 4. Scaled optimal outputs at grade transition – mass ofpolymer in reactor ys, reaction rate yr and hydrogen-ethylene ratio yhe.
grade change for all three reactors using up to 12 differentinflows has been done. Future work include reactor modelcalibration using measurements and modeling of distilla-tion towers for recycling of outflows.
REFERENCES
Akesson, J. (2008). Optimica—An Extension of ModelicaSupporting Dynamic Optimization. In In 6th Interna-tional Modelica Conference 2008. Modelica Association.
Akesson, J., Bergdahl, T., Gafvert, M., and Tummescheit,H. (2009). Modeling and Optimization with Modelicaand Optimica Using the JModelica.org Open SourcePlatform. In Proceedings of the 7th International Mod-elica Conference 2009. Modelica Association.
Biegler, L., Cervantes, A., and Wachter, A. (2002). Ad-vances in simultaneous strategies for dynamic optimiza-tion. Chemical Engineering Science, 57, 575–593.
van Brempt, W., van Overschee, P., Backx, T., and Moen,O. (2004). Plantwide economical dynamic optimization:Application on a Borealis Borstar process model. InADCHEM2003. Hong Kong.
